While the national Mastermath program allows you to follow courses all across the country, the master thesis project should make the difference in your choice of university. Have a look at a list of exciting thesis topics that you could explore at the Vrije Universiteit Amsterdam.
Project: Studying knots formed viral DNA
Supervisor: Senja Barthel
Short Description: The formation of knot types that appear in viral DNA are studied, allowing to test models of DNA packing. The knots can also be studied in their own right.
Project: Periodic orbits of relativistic particles
Supervisor: Gabriele Benedetti
Short description: This project studies the periodic motions of a charged particle hit by an electromagnetic wave in the presence of symmetry. We investigate the number of such motions and their relation with the underlying geometry of the space.
Project: Tight asymptotic bounds in Topological Data analysis
Supervisor: M.B. Botnan
Short Description: Persistent homology is a central tool in topological data analysis which offers a multiscale view of data. The standard algorithm runs in cubic time (input size is the number of simplices) and it is easy to see that the algorithm takes no more than cubic time. However, an example by Morozov shows that this bound is asymptotically tight. In this project, we shall study Morozov’s construction to construct new examples in single and multi-parameter persistent homology.
Project: Unveiling the connections of Cluster Algebras
Supervisor: İlke Çanakçı
Short Description: This master’s project delves into the rich and rapidly evolving field of Cluster Algebras, a branch of algebraic combinatorics introduced by Fomin and Zelevinsky. Cluster algebras have deep connections with areas like combinatorics, surface topology, and representation theory. The focus of the project may be on one of these themes, depending on the interest of the candidate, with the aim of exploring the role of cluster algebras in unifying various mathematical structures. Through this focused approach, the project seeks to deepen understanding of the structural properties and applications of cluster algebras.
Project: Modular forms, elliptic curves, and Diophantine equations
Supervisor: Sander Dahmen
Short description: Solving Diophantine equations, i.e. determining integer solutions to polynomial equations, is one of the oldest branches of mathematics, but still rapidly evolving. In this project, we will apply advanced tools from arithmetic geometry, most notably elliptic curve and modular forms, to tackle certain intriguing classes of Diophantine equations.
Project: Generalizing symplectic geometry from 1 to 2 dimensions
Supervisor: Oliver Fabert
Short description: Geodesics (= locally shortest paths) on Riemannian manifolds can be studied in the framework of symplectic geometry. Generalizing from one-dimensional geodesics to two-dimensional minimal surfaces (= soap films), it is the goal of this thesis to explore the corresponding two-dimensional generalization of symplectic geometry.
Project: p-adic geometry and the Langlands programme.
Supervisor: Pol van Hoften
Short description: Since the introduction of by Scholze in 2011, the theory of perfectoid spaces has revolutionized p-adic analytic geometry and modern algebraic number theory. There are a variety of possible projects in this direction that I could supervise, with topics such as condensed mathematics, (local) Shimura varieties, etc..
Project: Geometry and arithmetic
Supervisor: Rob de Jeu
Short description: Through the decades, many techniques have been developed that apply to both fields, leading to a more unified point of view. In this spectrum, many topics in number theory and/or geometry are possible (Brauer groups, elliptic curves, L-functions, models of curves, étale cohomology, …), and I would be happy to discuss them in more detail.
Project: Eilenberg–MacLane spaces via configuration spaces
Supervisor: Inbar Klang
Short description: Spaces of configurations of points in a manifold show up in many fields of mathematics, including algebraic topology, knot theory, and mathematical physics. For example, they can be used to model Eilenberg–MacLane spaces, which represent cohomology. This project aims to investigate how structures on cohomology manifest in the configuration space models of Eilenberg–MacLane spaces.
Project: Turán problem and eigenvalues of hypergraphs
Supervisor: Raffaella Mulas
Short description: Given integers n>= k > r >= 2, the Turán problem consists of determining or estimating the largest integer t such that there exists an r-uniform hypergraph on n vertices and t edges that does not contain any complete r-uniform hypergraph on k vertices as a sub-hypergraph. In this project, we aim to investigate the spectral properties of optimal solutions to this problem and its variants.
Project: Higher orientability
Supervisor: Thomas Rot
Short Description: A manifold is orientable if it is possible to consistently choose an orientation of the tangent bundle at each point. An example of an orientable manifold is the two dimensional sphere, while the Mobius strip is a non-orientable manifold. There are generalizations of this concept such as spin, string, 5-brane. These concepts are well studied in the classical literature. Recently Renee Hoekzema introduced a different kind of higher orientability. For this Hoekzema introduced a new class of spaces, whose topological properties will be studied in this project.